Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Review Article
  • Published:

Polarons in materials

A Publisher Correction to this article was published on 03 February 2022

An Author Correction to this article was published on 23 June 2021

This article has been updated

Abstract

Polarons are quasiparticles that easily form in polarizable materials due to the coupling of excess electrons or holes with ionic vibrations. These quasiparticles manifest themselves in many different ways and have a profound impact on materials properties and functionalities. Polarons have been the testing ground for the development of numerous theories, and their manifestations have been studied by many different experimental probes. This Review provides a map of the enormous amount of data and knowledge accumulated on polaron effects in materials, ranging from early studies and standard treatments to emerging experimental techniques and novel theoretical and computational approaches.

This is a preview of subscription content, access via your institution

Access options

Buy this article

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Basic properties of polarons.
Fig. 2: Polaron properties predicted by different theoretical and computational models.
Fig. 3: Interaction between two small polarons.
Fig. 4: Theoretical and experimental visualizations of different types of polarons.
Fig. 5: Transport measurements.
Fig. 6: Optical spectroscopies.
Fig. 7: ARPES, XPD and XAS measurements of large and small polarons.
Fig. 8: Real-space observations of polarons.
Fig. 9: Polarons in rutile and anatase TiO2.

Similar content being viewed by others

Change history

References

  1. Alexandrov, A. S. & Devreese, J. T. Advances in Polaron Physics (Springer, 2010).

  2. Bredas, J. L. & Street, G. B. Polarons, bipolarons, and solitons in conducting polymers. Acc. Chem. Res. 18, 309–315 (1985).

    Article  CAS  Google Scholar 

  3. Reticcioli, M., Diebold, U., Kresse, G. & Franchini, C. in Handbook of Materials Modeling (eds Andreoni, W. & Yip, S.) 1–39 (Springer, 2019).

  4. Pekar, S. I. Local quantum states of electrons in an ideal ion crystal. Zh. Eksp. Teor. Fiz. 16, 341–348 (1946).

    CAS  Google Scholar 

  5. Dykman, M. & Rashba, E. The roots of polaron theory. Phys. Today 68, 10 (2015).

    Article  Google Scholar 

  6. Landau, L. D. Über die bewegung der elektronen im kristallgitter. Phys. Z. Sowjetunion 3, 664–645 (1933).

    CAS  Google Scholar 

  7. Fröhlich, H., Pelzer, H. & Zienau, S. XX. Properties of slow electrons in polar materials. Lond. Edinb. Dubl. Philos. Mag. J. Sci. 41, 221–242 (1950).

    Article  Google Scholar 

  8. Frölich, H. Electrons in lattice fields. Adv. Phys. 3, 325–361 (1954).

    Article  Google Scholar 

  9. Holstein, T. Studies of polaron motion: Part I. The molecular-crystal model. Ann. Phys. 8, 325–342 (1959).

    Article  CAS  Google Scholar 

  10. Holstein, T. Studies of polaron motion: Part II. The “small” polaron. Ann. Phys. 8, 343–389 (1959).

    Article  CAS  Google Scholar 

  11. Feynman, R. P. Slow electrons in a polar crystal. Phys. Rev. 97, 660–665 (1955).

    Article  CAS  Google Scholar 

  12. Ōsaka, Y. Polaron state at a finite temperature. Prog. Theor. Phys. 22, 437–446 (1959).

    Article  Google Scholar 

  13. Sio, W. H., Verdi, C., Poncé, S. & Giustino, F. Ab initio theory of polarons: formalism and applications. Phys. Rev. B 99, 235139 (2019).

    Article  CAS  Google Scholar 

  14. Alexandrov, A. S. Polarons in Advanced Materials (Springer, 2007).

  15. Nagels, P., Denayer, M. & Devreese, J. Electrical properties of single crystals of uranium dioxide. Solid State Commun. 1, 35–40 (1963).

    Article  CAS  Google Scholar 

  16. Crevecoeur, C. & Wit, H. D. Electrical conductivity of Li doped MnO. J. Phys. Chem. Solids 31, 783–791 (1970).

    Article  CAS  Google Scholar 

  17. Stoneham, A. M. et al. Trapping, self-trapping and the polaron family. J. Phys. Condens. Matter 19, 255208 (2007).

    Article  Google Scholar 

  18. Coropceanu, V. et al. Charge transport in organic semiconductors. Chem. Rev. 107, 926–952 (2007).

    Article  CAS  Google Scholar 

  19. Zhugayevych, A. & Tretiak, S. Theoretical description of structural and electronic properties of organic photovoltaic materials. Annu. Rev. Phys. Chem. 66, 305–330 (2015).

    Article  CAS  Google Scholar 

  20. Roth, S. & Carroll, D. in Foundations of Solid State Physics: Dimensionality and Symmetry Ch. 9 (eds Roth, S. & Carroll, D.) 301–401 (Wiley, 2019).

  21. Cobet, C., Gasiorowski, J., Farka, D. & Stadler, P. in Ellipsometry of Functional Organic Surfaces and Films (eds Hinrichs, K. & Eichhorn, K. J.) 355–387 (Springer, 2018).

  22. De Sio, A. et al. Tracking the coherent generation of polaron pairs in conjugated polymers. Nat. Commun. 7, 13742 (2016).

    Article  Google Scholar 

  23. Kaminski, A. & Das Sarma, S. Polaron percolation in diluted magnetic semiconductors. Phys. Rev. Lett. 88, 247202 (2002).

    Article  CAS  Google Scholar 

  24. Teresa, J. M. D. et al. Evidence for magnetic polarons in the magnetoresistive perovskites. Nature 386, 256–259 (1997).

    Article  Google Scholar 

  25. Zhou, J.-S. & Goodenough, J. B. Zener versus de Gennes ferromagnetism in La1−xSrxMnO3. Phys. Rev. B 62, 3834–3838 (2000).

    Article  CAS  Google Scholar 

  26. Daoud-Aladine, A., Rodriguez-Carvajal, J., Pinsard-Gaudart, L., Fernández-Díaz, M. T. & Revcolevschi, A. Zener polaron ordering in half-doped manganites. Phys. Rev. Lett. 89, 097205 (2002).

    Article  CAS  Google Scholar 

  27. Yamada, Y. et al. Polaron ordering in low-doping La1−xSrxMnO3. Phys. Rev. Lett. 77, 904–907 (1996).

    Article  CAS  Google Scholar 

  28. Zhao, G.-M., Hunt, M. B., Keller, H. & Müller, K. A. Evidence for polaronic supercarriers in the copper oxide superconductors La2−xSrxCuO4. Nature 385, 236–239 (1997).

    Article  CAS  Google Scholar 

  29. Cortecchia, D. et al. Polaron self-localization in white-light emitting hybrid perovskites. J. Mater. Chem. C 5, 2771–2780 (2017).

    Article  CAS  Google Scholar 

  30. Miyata, K. et al. Large polarons in lead halide perovskites. Sci. Adv. 3, e1701217 (2017).

    Article  Google Scholar 

  31. Chen, Q., Wang, W. & Peeters, F. M. Magneto-polarons in monolayer transition-metal dichalcogenides. J. Appl. Phys. 123, 214303 (2018).

    Article  Google Scholar 

  32. Kang, M. et al. Holstein polaron in a valley-degenerate two-dimensional semiconductor. Nat. Mater. 17, 676–680 (2018).

    Article  CAS  Google Scholar 

  33. McKenna, K. P., Wolf, M. J., Shluger, A. L., Lany, S. & Zunger, A. Two-dimensional polaronic behavior in the binary oxides m−HfO2 and m−ZrO2. Phys. Rev. Lett. 108, 116403 (2012).

    Article  Google Scholar 

  34. Natanzon, Y., Azulay, A. & Amouyal, Y. Evaluation of polaron transport in solids from first-principles. Isr. J. Chem. 60, 768–786 (2020).

    Article  CAS  Google Scholar 

  35. Nelson, J., Kwiatkowski, J. J., Kirkpatrick, J. & Frost, J. M. Modeling charge transport in organic photovoltaic materials. Acc. Chem. Res. 42, 1768–1778 (2009).

    Article  CAS  Google Scholar 

  36. Ortmann, F., Bechstedt, F. & Hannewald, K. Charge transport in organic crystals: theory and modelling. Phys. Status Solidi B 248, 511–525 (2011).

    Article  CAS  Google Scholar 

  37. Di Valentin, C., Pacchioni, G. & Selloni, A. Reduced and n-type doped TiO2: Nature of Ti3+ species. J. Phys. Chem. C 113, 20543–20552 (2009).

    Article  Google Scholar 

  38. Papageorgiou, A. C. et al. Electron traps and their effect on the surface chemistry of TiO2(110). Proc. Natl Acad. Sci. USA 107, 2391–2396 (2010).

    Article  Google Scholar 

  39. Reticcioli, M. et al. Interplay between adsorbates and polarons: CO on rutile TiO2(110). Phys. Rev. Lett. 122, 016805 (2019).

    Article  CAS  Google Scholar 

  40. Yin, W.-J., Wen, B., Zhou, C., Selloni, A. & Liu, L.-M. Excess electrons in reduced rutile and anatase TiO2. Surf. Sci. Rep. 73, 58–82 (2018).

    Article  CAS  Google Scholar 

  41. Reticcioli, M. et al. Polaron-driven surface reconstructions. Phys. Rev. X 7, 031053 (2017).

    Google Scholar 

  42. Millis, A. J., Mueller, R. & Shraiman, B. I. Fermi-liquid-to-polaron crossover. II. double exchange and the physics of colossal magnetoresistance. Phys. Rev. B 54, 5405–5417 (1996).

    Article  CAS  Google Scholar 

  43. Wang, M. et al. Thermoelectric Seebeck effect in oxide-based resistive switching memory. Nat. Commun. 5, 4598 (2014).

    Article  CAS  Google Scholar 

  44. Verdi, C., Caruso, F. & Giustino, F. Origin of the crossover from polarons to fermi liquids in transition metal oxides. Nat. Commun. 8, 15769 (2017).

    Article  CAS  Google Scholar 

  45. Miyata, K. & Zhu, X.-Y. Ferroelectric large polarons. Nat. Mater. 17, 379–381 (2018).

    Article  CAS  Google Scholar 

  46. Setvin, M. et al. Direct view at excess electrons in TiO2 rutile and anatase. Phys. Rev. Lett. 113, 086402 (2014).

    Article  CAS  Google Scholar 

  47. Dehn, M. H. et al. Observation of a charge-neutral muon-polaron complex in antiferromagnetic Cr2O3. Phys. Rev. X 10, 011036 (2020).

    CAS  Google Scholar 

  48. Moser, S. et al. Tunable polaronic conduction in anatase TiO2. Phys. Rev. Lett. 110, 196403 (2013).

    Article  CAS  Google Scholar 

  49. de Groot, F. & Kotani, A. Core Level Spectroscopy of Solids (CRC, 2008).

  50. Di Valentin, C., Pacchioni, G., Selloni, A., Livraghi, S. & Giamello, E. Characterization of paramagnetic species in N-doped TiO2 powders by EPR spectroscopy and DFT calculations. J. Phys. Chem. B 109, 11414–11419 (2005).

    Article  Google Scholar 

  51. Yang, S., Brant, A. T., Giles, N. C. & Halliburton, L. E. Intrinsic small polarons in rutile TiO2. Phys. Rev. B 87, 125201 (2013).

    Article  Google Scholar 

  52. Rho, H. et al. Evolution of magnetic polarons and spin-carrier interactions through the metal-insulator transition in Eu1−xGdxO. Phys. Rev. Lett. 88, 127401 (2002).

    Article  CAS  Google Scholar 

  53. Storchak, V. G. et al. Magnetic polarons in Eu-based films of magnetic semiconductors. Phys. Rev. B 81, 153201 (2010).

    Article  Google Scholar 

  54. Matus, M., Kuzmany, H. & Sohmen, E. Self-trapped polaron exciton in neutral fullerene C60. Phys. Rev. Lett. 68, 2822–2825 (1992).

    Article  CAS  Google Scholar 

  55. Sezen, H. et al. Probing electrons in TiO2 polaronic trap states by IR-absorption: evidence for the existence of hydrogenic states. Sci. Rep. 4, 3808 (2014).

    Article  Google Scholar 

  56. Devreese, J. T., Klimin, S. N., van Mechelen, J. L. M. & van der Marel, D. Many-body large polaron optical conductivity in SrTi1−xNbxO3. Phys. Rev. B 81, 1252–1259 (2010).

    Article  Google Scholar 

  57. Klimin, S., Tempere, J., Devreese, J. T., Franchini, C. & Kresse, G. Optical response of an interacting polaron gas in strongly polar crystals. Appl. Sci. 10, 2059 (2020).

    Article  CAS  Google Scholar 

  58. Prokof’ev, N. V. & Svistunov, B. V. Polaron problem by diagrammatic quantum Monte Carlo. Phys. Rev. Lett. 81, 2514–2517 (1998).

    Article  Google Scholar 

  59. Mishchenko, A. S., Prokof’ev, N. V., Sakamoto, A. & Svistunov, B. V. Diagrammatic quantum Monte Carlo study of the Fröhlich polaron. Phys. Rev. B 62, 6317–6336 (2000).

    Article  CAS  Google Scholar 

  60. Titantah, J. T., Pierleoni, C. & Ciuchi, S. Free energy of the Fröhlich polaron in two and three dimensions. Phys. Rev. Lett. 87, 206406 (2001).

    Article  CAS  Google Scholar 

  61. Bredow, T. & Pacchioni, G. Electronic structure of an isolated oxygen vacancy at the TiO2(110) surface. Chem. Phys. Lett. 355, 417–423 (2002).

    Article  CAS  Google Scholar 

  62. Lany, S. & Zunger, A. Polaronic hole localization and multiple hole binding of acceptors in oxide wide-gap semiconductors. Phys. Rev. B 80, 085202 (2009).

    Article  Google Scholar 

  63. Kokott, S., Levchenko, S. V., Rinke, P. & Scheffler, M. First-principles supercell calculations of small polarons with proper account for long-range polarization effects. New J. Phys. 20, 033023 (2018).

    Article  Google Scholar 

  64. Verdi, C. & Giustino, F. Fröhlich electron-phonon vertex from first principles. Phys. Rev. Lett. 115, 176401 (2015).

    Article  Google Scholar 

  65. Ciuchi, S., de Pasquale, F. & Feinberg, D. Exact solution of the small-polaron problem in infinite dimensions. EPL 30, 151 (1995).

    Article  CAS  Google Scholar 

  66. Ciuchi, S., de Pasquale, F., Fratini, S. & Feinberg, D. Dynamical mean-field theory of the small polaron. Phys. Rev. B 56, 4494–4512 (1997).

    Article  CAS  Google Scholar 

  67. Fratini, S. & Ciuchi, S. Dynamical mean-field theory of transport of small polarons. Phys. Rev. Lett. 91, 256403 (2003).

    Article  CAS  Google Scholar 

  68. Kerisit, S. & Rosso, K. M. Kinetic Monte Carlo model of charge transport in hematite (α−Fe2O3). J. Chem. Phys. 127, 124706 (2007).

    Article  Google Scholar 

  69. Wolf, M. J., Irvine, L. A. D. & Walker, A. B. Quantifying polaronic effects on charge-carrier scattering and mobility in lead–halide perovskite. Preprint at arXiv https://arxiv.org/abs/2003.00968 (2020).

  70. Sio, W. H., Verdi, C., Poncé, S. & Giustino, F. Polarons from first principles, without supercells. Phys. Rev. Lett. 122, 246403 (2019).

    Article  CAS  Google Scholar 

  71. Gono, P., Wiktor, J., Ambrosio, F. & Pasquarello, A. Surface polarons reducing overpotentials in the oxygen evolution reaction. ACS Catal. 8, 5847–5851 (2018).

    Article  CAS  Google Scholar 

  72. Zhang, D., Han, Z.-K., Murgida, G. E., Ganduglia-Pirovano, M. V. & Gao, Y. Oxygen-vacancy dynamics and entanglement with polaron hopping at the reduced CeO2(111) surface. Phys. Rev. Lett. 122, 096101 (2019).

    Article  CAS  Google Scholar 

  73. Bombile, J. H., Janik, M. J. & Milner, S. T. Polaron formation mechanisms in conjugated polymers. Phys. Chem. Chem. Phys. 20, 317–331 (2018).

    Article  CAS  Google Scholar 

  74. Yuan, S., Wang, Z., Baron, M. L. F. & Bevan, K. H. Ab initio insight into the formation of small polarons: A study across four metal peroxides. Phys. Rev. B 100, 205201 (2019).

    Article  CAS  Google Scholar 

  75. Gerosa, M., Gygi, F., Govoni, M. & Galli, G. The role of defects and excess surface charges at finite temperature for optimizing oxide photoabsorbers. Nat. Mater. 17, 1122–1127 (2018).

    Article  CAS  Google Scholar 

  76. Nery, J. P. et al. Quasiparticles and phonon satellites in spectral functions of semiconductors and insulators: Cumulants applied to the full first-principles theory and the Fröhlich polaron. Phys. Rev. B 97, 115145 (2018).

    Article  CAS  Google Scholar 

  77. Swartz, A. G. et al. Polaronic behavior in a weak-coupling superconductor. Proc. Natl Acad. Sci. USA 115, 1475–1480 (2018).

    Article  CAS  Google Scholar 

  78. Mora-Fonz, D. & Shluger, A. L. Modeling of intrinsic electron and hole trapping in crystalline and amorphous ZnO. Adv. Electron. Mater. 6, 1900760 (2020).

    Article  CAS  Google Scholar 

  79. Huang, H. Y. et al. Jahn-Teller distortion driven magnetic polarons in magnetite. Nat. Commun. 8, 15929 (2017).

    Article  CAS  Google Scholar 

  80. Zhou, J.-J. & Bernardi, M. Predicting charge transport in the presence of polarons: The beyond-quasiparticle regime in SrTiO3. Phys. Rev. Res. 1, 033138 (2019).

    Article  CAS  Google Scholar 

  81. Mishchenko, A. S. et al. Polaron mobility in the “beyond quasiparticles” regime. Phys. Rev. Lett. 123, 076601 (2019).

    Article  CAS  Google Scholar 

  82. Zhang, S. et al. Enhanced superconducting state in FeSe/SrTiO3 by a dynamic interfacial polaron mechanism. Phys. Rev. Lett. 122, 066802 (2019).

    Article  CAS  Google Scholar 

  83. Ismail, A. S. M. et al. Direct observation of the electronic states of photoexcited hematite with ultrafast 2p3d X-ray absorption spectroscopy and resonant inelastic X-ray scattering. Phys. Chem. Chem. Phys. 22, 2685–2692 (2020).

    Article  CAS  Google Scholar 

  84. Pastor, E. et al. In situ observation of picosecond polaron self-localisation in α-Fe2O3 photoelectrochemical cells. Nat. Commun. 10, 3962 (2019).

    Article  Google Scholar 

  85. Lee, T. D., Low, F. E. & Pines, D. The motion of slow electrons in a polar crystal. Phys. Rev. 90, 297–302 (1953).

    Article  Google Scholar 

  86. Zienau, S. ERA report L/T236 (ILC, 1950).

  87. Lang, I. G. & Firsov, Y. A. Kinetic theory of semiconductors with low mobility. Sov. Phys. JETP 16, 1301 (1962).

    Google Scholar 

  88. Rongsheng, H., Zijing, L. & Kelin, W. Exact solutions for the two-site Holstein model. Phys. Rev. B 65, 174303 (2002).

    Article  Google Scholar 

  89. Tayebi, A. & Zelevinsky, V. The Holstein polaron problem revisited. J. Phys. A Math. Theor. 49, 255004 (2016).

    Article  Google Scholar 

  90. Hohenadler, M., Evertz, H. G. & von der Linden, W. Quantum Monte Carlo and variational approaches to the Holstein model. Phys. Rev. B 69, 024301 (2004).

    Article  Google Scholar 

  91. Hahn, T., Klimin, S., Tempere, J., Devreese, J. T. & Franchini, C. Diagrammatic Monte Carlo study of Fröhlich polaron dispersion in two and three dimensions. Phys. Rev. B 97, 134305 (2018).

    Article  CAS  Google Scholar 

  92. Goodvin, G. L., Berciu, M. & Sawatzky, G. A. Green’s function of the Holstein polaron. Phys. Rev. B 74, 245104 (2006).

    Article  Google Scholar 

  93. Rosenfelder, R. & Schreiber, A. On the best quadratic approximation in Feynman’s path integral treatment of the polaron. Phys. Lett. A 284, 63–71 (2001).

    Article  CAS  Google Scholar 

  94. Becker, W., Gerlach, B. & Schliffke, H. Monte Carlo calculation of the ground-state energy of an optical polaron. Phys. Rev. B 28, 5735–5738 (1983).

    Article  Google Scholar 

  95. Gelfand, J. M. & Chentsov, N. N. The numerical calculation of path integrals. Zh. Eksp. Teor. Fiz. 3, 1106 (1957).

    Google Scholar 

  96. Sabelfeld, K. K. Approximate evaluation of wiener continual integrals by the Monte Carlo method. USSR Computational Math. Phys. 19, 27–43 (1979).

    Article  Google Scholar 

  97. Mishchenko, A. S., Nagaosa, N. & Prokof’ev, N. Diagrammatic Monte Carlo method for many-polaron problems. Phys. Rev. Lett. 113, 166402 (2014).

    Article  Google Scholar 

  98. Van Houcke, K., Kozik, E., Prokof’ev, N. & Svistunov, B. Diagrammatic Monte Carlo. Phys. Procedia 6, 95–105 (2010).

    Article  Google Scholar 

  99. Mishchenko, A. S. Diagrammatic Monte Carlo method as applied to the polaron problems. Phys. Uspekhi 48, 887 (2005).

    Article  CAS  Google Scholar 

  100. Mishchenko, A. S., De Filippis, G., Cataudella, V., Nagaosa, N. & Fehske, H. Optical signatures of exciton polarons from diagrammatic Monte Carlo. Phys. Rev. B 97, 045141 (2018).

    Article  CAS  Google Scholar 

  101. Alexandrov, A. S., Kabanov, V. V. & Ray, D. K. From electron to small polaron: an exact cluster solution. Phys. Rev. B 49, 9915–9923 (1994).

    Article  CAS  Google Scholar 

  102. Grusdt, F. All-coupling theory for the Fröhlich polaron. Phys. Rev. B 93, 144302 (2016).

    Article  Google Scholar 

  103. Lemmens, L. F., Brosens, F. & Devreese, J. T. On the ground state energy of a gas of interacting polarons. Phys. Status Solidi B 82, 439–447 (1977).

    Article  CAS  Google Scholar 

  104. Tempere, J. & Devreese, J. T. Optical absorption of an interacting many-polaron gas. Phys. Rev. B 64, 104504 (2001).

    Article  Google Scholar 

  105. Hedin, L. New method for calculating the one-particle Green’s function with application to the electron-gas problem. Phys. Rev. 139, A796–A823 (1965).

    Article  Google Scholar 

  106. Ergönenc, Z., Kim, B., Liu, P., Kresse, G. & Franchini, C. Converged GW quasiparticle energies for transition metal oxide perovskites. Phys. Rev. Mater. 2, 024601 (2018).

    Article  Google Scholar 

  107. Becke, A. D. A new mixing of Hartree–Fock and local density-functional theories. J. Chem. Phys. 98, 1372–1377 (1993).

    Article  CAS  Google Scholar 

  108. Franchini, C. Hybrid functionals applied to perovskites. J. Phys. Condens. Matter 26, 253202 (2014).

    Article  Google Scholar 

  109. Anisimov, V. I., Zaanen, J. & Andersen, O. K. Band theory and Mott insulators: Hubbard U instead of Stoner I. Phys. Rev. B 44, 943–954 (1991).

    Article  CAS  Google Scholar 

  110. Himmetoglu, B., Floris, A., de Gironcoli, S. & Cococcioni, M. Hubbard-corrected DFT energy functionals: The LDA+U description of correlated systems. Int. J. Quantum Chem. 114, 14–49 (2014).

    Article  CAS  Google Scholar 

  111. Baroni, S., de Gironcoli, S., Dal Corso, A. & Giannozzi, P. Phonons and related crystal properties from density-functional perturbation theory. Rev. Mod. Phys. 73, 515–562 (2001).

    Article  CAS  Google Scholar 

  112. Giustino, F. Electron-phonon interactions from first principles. Rev. Mod. Phys. 89, 015003 (2017).

    Article  Google Scholar 

  113. Kohn, W. & Sham, L. J. Self-consistent equations including exchange and correlation effects. Phys. Rev. 140, A1133–A1138 (1965).

    Article  Google Scholar 

  114. Lejaeghere, K. et al. Reproducibility in density functional theory calculations of solids. Science 351, aad3000 (2016).

    Article  Google Scholar 

  115. Belviso, F. et al. Viewpoint: Atomic-scale design protocols toward energy, electronic, catalysis, and sensing applications. Inorg. Chem. 58, 14939–14980 (2019).

    Article  CAS  Google Scholar 

  116. Seidl, A., Görling, A., Vogl, P., Majewski, J. A. & Levy, M. Generalized Kohn-Sham schemes and the band-gap problem. Phys. Rev. B 53, 3764–3774 (1996).

    Article  CAS  Google Scholar 

  117. Nolan, M. & Watson, G. W. Hole localization in Al doped silica: A DFT+U description. J. Chem. Phys. 125, 144701 (2006).

    Article  Google Scholar 

  118. Maxisch, T., Zhou, F. & Ceder, G. Ab initio study of the migration of small polarons in olivine LixFePO4 and their association with lithium ions and vacancies. Phys. Rev. B 73, 104301 (2006).

    Article  Google Scholar 

  119. Marcus, R. A. Electron transfer reactions in chemistry. Theory and experiment. Rev. Mod. Phys. 65, 599–610 (1993).

    Article  CAS  Google Scholar 

  120. Marcus, R. A. Chemical and electrochemical electron-transfer theory. Annu. Rev. Phys. Chem. 15, 155–196 (1964).

    Article  CAS  Google Scholar 

  121. Deskins, N. A. & Dupuis, M. Electron transport via polaron hopping in bulk TiO2: A density functional theory characterization. Phys. Rev. B 75, 195212 (2007).

    Article  Google Scholar 

  122. Sun, L., Huang, X., Wang, L. & Janotti, A. Disentangling the role of small polarons and oxygen vacancies in CeO2. Phys. Rev. B 95, 245101 (2017).

    Article  Google Scholar 

  123. Castleton, C. W. M., Lee, A. & Kullgren, J. Benchmarking density functional theory functionals for polarons in oxides: properties of CeO2. J. Phys. Chem. C 123, 5164–5175 (2019).

    Article  CAS  Google Scholar 

  124. Wang, Z., Brock, C., Matt, A. & Bevan, K. H. Implications of the DFT+u method on polaron properties in energy materials. Phys. Rev. B 96, 125150 (2017).

    Article  Google Scholar 

  125. Kick, M., Reuter, K. & Oberhofer, H. Intricacies of DFT+U, not only in a numeric atom centered orbital framework. J. Chem. Theory Comput. 15, 1705–1718 (2019).

    Article  CAS  Google Scholar 

  126. Cococcioni, M. & de Gironcoli, S. Linear response approach to the calculation of the effective interaction parameters in the LDA+U method. Phys. Rev. B 71, 035105 (2005).

    Article  Google Scholar 

  127. Aryasetiawan, F., Karlsson, K., Jepsen, O. & Schönberger, U. Calculations of Hubbard U from first-principles. Phys. Rev. B 74, 125106 (2006).

    Article  Google Scholar 

  128. Perdew, J. P. & Zunger, A. Self-interaction correction to density-functional approximations for many-electron systems. Phys. Rev. B 23, 5048–5079 (1981).

    Article  CAS  Google Scholar 

  129. Janesko, B. G., Henderson, T. M. & Scuseria, G. E. Screened hybrid density functionals for solid-state chemistry and physics. Phys. Chem. Chem. Phys. 11, 443–454 (2009).

    Article  CAS  Google Scholar 

  130. Pacchioni, G., Frigoli, F., Ricci, D. & Weil, J. A. Theoretical description of hole localization in a quartz Al center: The importance of exact electron exchange. Phys. Rev. B 63, 054102 (2000).

    Article  Google Scholar 

  131. Bjaalie, L. et al. Small hole polarons in rare-earth titanates. Appl. Phys. Lett. 106, 232103 (2015).

    Article  Google Scholar 

  132. Lany, S. Predicting polaronic defect states by means of generalized Koopmans density functional calculations. Phys. Status Solidi B 248, 1052–1060 (2011).

    Article  CAS  Google Scholar 

  133. Tabriz, M. F., Aradi, B., Frauenheim, T. & Deák, P. Application of the Lany–Zunger polaron correction for calculating surface charge trapping. J. Phys. Condens. Matter 29, 394001 (2017).

    Article  Google Scholar 

  134. Elmaslmane, A. R., Wetherell, J., Hodgson, M. J., McKenna, K. P. & Godby, R. W. Accuracy of electron densities obtained via Koopmans-compliant hybrid functionals. Phys. Rev. Mater. 2, 040801 (2018).

    Article  CAS  Google Scholar 

  135. Elmaslmane, A. R., Watkins, M. B. & McKenna, K. P. First-principles modeling of polaron formation in TiO2 polymorphs. J. Chem. Theory Comput. 14, 3740–3751 (2018).

    Article  CAS  Google Scholar 

  136. Makov, G. & Payne, M. C. Periodic boundary conditions in ab initio calculations. Phys. Rev. B 51, 4014–4022 (1995).

    Article  CAS  Google Scholar 

  137. Shluger, A. L. & Stoneham, A. M. Small polarons in real crystals: concepts and problems. J. Phys. Condens. Matter 5, 3049–3086 (1993).

    Article  CAS  Google Scholar 

  138. Pinto, H. & Stashans, A. Computational study of self-trapped hole polarons in tetragonal BaTiO3. Phys. Rev. B 65, 134304 (2002).

    Article  Google Scholar 

  139. Eglitis, R., Eglitis, R., Kotomin, E., Kotomin, E. & Borstel, G. Semi-empirical calculations of hole polarons in MgO and KNbO3 crystals. Phys. Status Solidi B 208, 15–20 (1998).

    Article  CAS  Google Scholar 

  140. Neukirch, A. J. et al. Polaron stabilization by cooperative lattice distortion and cation rotations in hybrid perovskite materials. Nano Lett. 16, 3809–3816 (2016).

    Article  CAS  Google Scholar 

  141. Kick, M. & Oberhofer, H. Towards a transferable design of solid-state embedding models on the example of a rutile TiO2 (110) surface. J. Chem. Phys. 151, 184114 (2019).

    Article  CAS  Google Scholar 

  142. Berger, D., Oberhofer, H. & Reuter, K. First-principles embedded-cluster calculations of the neutral and charged oxygen vacancy at the rutile TiO2(110) surface. Phys. Rev. B 92, 075308 (2015).

    Article  Google Scholar 

  143. Ghosh, D., Welch, E., Neukirch, A. J., Zakhidov, A. & Tretiak, S. Polarons in halide perovskites: a perspective. J. Phys. Chem. Lett. 11, 3271–3286 (2020).

    Article  CAS  Google Scholar 

  144. Mishchenko, A. S., Nagaosa, N., De Filippis, G., de Candia, A. & Cataudella, V. Mobility of Holstein polaron at finite temperature: an unbiased approach. Phys. Rev. Lett. 114, 146401 (2015).

    Article  CAS  Google Scholar 

  145. Emin, D. Polarons (Cambridge Univ. Press, 2012).

  146. Brunschwig, B. S., Logan, J., Newton, M. D. & Sutin, N. A semiclassical treatment of electron-exchange reactions. Application to the hexaaquoiron(II)-hexaaquoiron(III) system. J. Am. Chem. Soc. 102, 5798–5809 (1980).

    Article  CAS  Google Scholar 

  147. Marcus, R. A. & Sutin, N. Electron transfers in chemistry and biology. Biochim. Biophys. Acta 811, 265–322 (1985).

    Article  CAS  Google Scholar 

  148. Emin, D. & Holstein, T. Studies of small-polaron motion IV. Adiabatic theory of the Hall effect. Ann. Phys. 53, 439–520 (1969).

    Article  Google Scholar 

  149. Holstein, T. Studies of polaron motion: Part II. The “small” polaron. Ann. Phys. 281, 725–773 (2000).

    Article  CAS  Google Scholar 

  150. Austin, I. G. & Mott, N. F. Polarons in crystalline and non-crystalline materials. Adv. Phys. 50, 757–812 (2001).

    Article  Google Scholar 

  151. Spreafico, C. & VandeVondele, J. The nature of excess electrons in anatase and rutile from hybrid DFT and RPA. Phys. Chem. Chem. Phys. 16, 26144–26152 (2014).

    Article  CAS  Google Scholar 

  152. Bondarenko, N., Eriksson, O. & Skorodumova, N. V. Polaron mobility in oxygen-deficient and lithium-doped tungsten trioxide. Phys. Rev. B 92, 165119 (2015).

    Article  Google Scholar 

  153. Janotti, A., Franchini, C., Varley, J. B., Kresse, G. & Van de Walle, C. G. Dual behavior of excess electrons in rutile TiO2. Phys. Status Solidi Rapid Res. Lett. 7, 199–203 (2013).

    Article  CAS  Google Scholar 

  154. Kowalski, P. M., Camellone, M. F., Nair, N. N., Meyer, B. & Marx, D. Charge localization dynamics induced by oxygen vacancies on the TiO2(110) surface. Phys. Rev. Lett. 105, 146405 (2010).

    Article  Google Scholar 

  155. Reticcioli, M., Setvin, M., Schmid, M., Diebold, U. & Franchini, C. Formation and dynamics of small polarons on the rutile TiO2(110) surface. Phys. Rev. B 98, 045306 (2018).

    Article  CAS  Google Scholar 

  156. Emin, S. I. Small polarons. Phys. Today 35, 34–40 (1982).

    Article  CAS  Google Scholar 

  157. Nelson, J. Continuous-time random-walk model of electron transport in nanocrystalline TiO2 electrodes. Phys. Rev. B 59, 15374–15380 (1999).

    Article  CAS  Google Scholar 

  158. Kadanoff, L. P. Boltzmann equation for polarons. Phys. Rev. 130, 1364–1369 (1963).

    Article  Google Scholar 

  159. Motta, C. & Sanvito, S. Electron–phonon coupling and polaron mobility in hybrid perovskites from first principles. J. Phys. Chem. C 122, 1361–1366 (2018).

    Article  CAS  Google Scholar 

  160. Feynman, R. P., Hellwarth, R. W., Iddings, C. K. & Platzman, P. M. Mobility of slow electrons in a polar crystal. Phys. Rev. 127, 1004–1017 (1962).

    Article  Google Scholar 

  161. Hellwarth, R. W. & Biaggio, I. Mobility of an electron in a multimode polar lattice. Phys. Rev. B 60, 299–307 (1999).

    Article  CAS  Google Scholar 

  162. Frost, J. M. Calculating polaron mobility in halide perovskites. Phys. Rev. B 96, 195202 (2017).

    Article  Google Scholar 

  163. Gartstein, Y. Charges on semiconducting nanotubes in polar media: Polarons and excitons. Phys. Lett. A 349, 377–383 (2006).

    Article  CAS  Google Scholar 

  164. Basko, D. M. & Conwell, E. M. Effect of solvation on hole motion in DNA. Phys. Rev. Lett. 88, 098102 (2002).

    Article  CAS  Google Scholar 

  165. Setvin, M. et al. Charge trapping at the step edges of TiO2 anatase (101). Angew. Chem. Int. Ed. 53, 4714–4716 (2014).

    Article  CAS  Google Scholar 

  166. Di Valentin, C., Pacchioni, G. & Selloni, A. Electronic structure of defect states in hydroxylated and reduced rutile TiO2(110) surfaces. Phys. Rev. Lett. 97, 166803 (2006).

    Article  Google Scholar 

  167. Wiktor, J. & Pasquarello, A. Electron and hole polarons at the BiVO4–water interface. ACS Appl. Mater. Interfaces 11, 18423–18426 (2019).

    Article  CAS  Google Scholar 

  168. Selcuk, S. & Selloni, A. Facet-dependent trapping and dynamics of excess electrons at anatase TiO2 surfaces and aqueous interfaces. Nat. Mater. 15, 1107–1112 (2016).

    Article  CAS  Google Scholar 

  169. Salje, E. K. H., Alexandrov, A. S. & Liang, W. Y. Polarons and Bipolarons in High-Tc Superconductors and Related Materials (Cambridge Univ. Press, 1995).

  170. Salje, E. K. H. Polarons and bipolarons in tungsten oxide WO3−x. Eur. J. Solid State Inorg. Chem. 31, 805–821 (1994).

    CAS  Google Scholar 

  171. Williams, R. & Song, K. The self-trapped exciton. J. Phys. Chem. Solids 51, 679–716 (1990).

    Article  CAS  Google Scholar 

  172. Li, S., Luo, J., Liu, J. & Tang, J. Self-trapped excitons in all-inorganic halide perovskites: fundamentals, status, and potential applications. J. Phys. Chem. Lett. 10, 1999–2007 (2019).

    Article  CAS  Google Scholar 

  173. Srimath Kandada, A. R. & Silva, C. Exciton polarons in two-dimensional hybrid metal-halide perovskites. J. Phys. Chem. Lett. 11, 3173–3184 (2020).

    Article  CAS  Google Scholar 

  174. Zhu, X. et al. Charge transfer excitons at van der Waals interfaces. J. Am. Chem. Soc. 137, 8313–8320 (2015).

    Article  CAS  Google Scholar 

  175. Sangalli, D., Perfetto, E., Stefanucci, G. & Marini, A. An ab-initio approach to describe coherent and non-coherent exciton dynamics. Eur. Phys. J. B 91, 171 (2018).

    Article  Google Scholar 

  176. Xu, T. et al. Electron engineering of metallic multiferroic polarons in epitaxial BaTiO3. Npj Comput. Mater. 5, 23 (2019).

    Article  Google Scholar 

  177. Colizzi, G., Filippetti, A. & Fiorentini, V. Multiferroicity and orbital ordering in Pr0.5Ca0.5MnO3 from first principles. Phys. Rev. B 82, 140101 (2010).

    Article  Google Scholar 

  178. Soriano, D. & Katsnelson, M. I. Magnetic polaron and antiferromagnetic-ferromagnetic transition in doped bilayer CrI3. Phys. Rev. B 101, 041402 (2020).

    Article  CAS  Google Scholar 

  179. Kweon, K. E., Hwang, G. S., Kim, J., Kim, S. & Kim, S. Electron small polarons and their transport in bismuth vanadate: a first principles study. Phys. Chem. Chem. Phys. 17, 256–260 (2015).

    Article  CAS  Google Scholar 

  180. Ambrosio, F. & Wiktor, J. Strong hole trapping due to oxygen dimers in BiVO4: effect on the water oxidation reaction. J. Phys. Chem. Lett. 10, 7113–7118 (2019).

    Article  CAS  Google Scholar 

  181. Han, Z.-K., Zhang, L., Liu, M., Ganduglia-Pirovano, M. V. & Gao, Y. The structure of oxygen vacancies in the near-surface of reduced CeO2 (111) under strain. Front. Chem. 7, 436 (2019).

    Article  CAS  Google Scholar 

  182. Swift, M., Janotti, A. & Van de Walle, C. G. Small polarons and point defects in barium cerate. Phys. Rev. B 92, 214114 (2015).

    Article  Google Scholar 

  183. Bjaalie, L., Janotti, A., Krishnaswamy, K. & Van de Walle, C. G. Point defects, impurities, and small hole polarons in GdTiO3. Phys. Rev. B 93, 115316 (2016).

    Article  Google Scholar 

  184. Chen, C., Avila, J., Frantzeskakis, E., Levy, A. & Asensio, M. C. Observation of a two-dimensional liquid of Fröhlich polarons at the bare SrTiO3 surface. Nat. Commun. 6, 8585 (2015).

    Article  CAS  Google Scholar 

  185. Hauser, A. J., Mikheev, E., Kajdos, A. P. & Janotti, A. Small polaron-related recombination in BaxSr1−xTiO3 thin films by cathodoluminescence spectroscopy. Appl. Phys. Lett. 108, 102901 (2016).

    Article  Google Scholar 

  186. Himmetoglu, B., Janotti, A., Bjaalie, L. & Van de Walle, C. G. Interband and polaronic excitations in YTiO3 from first principles. Phys. Rev. B 90, 161102 (2014).

    Article  Google Scholar 

  187. Himmetoglu, B. & Janotti, A. Transport properties of KtaO3 from first-principles. J. Phys. Condens. Matter 28, 065502 (2016).

    Article  Google Scholar 

  188. Krishnaswamy, K., Himmetoglu, B., Kang, Y., Janotti, A. & Van de Walle, C. G. First-principles analysis of electron transport in BaSnO3. Phys. Rev. B 95, 205202 (2017).

    Article  Google Scholar 

  189. Körbel, S., Hlinka, J. & Sanvito, S. Electron trapping by neutral pristine ferroelectric domain walls in BiFeO3. Phys. Rev. B 98, 100104 (2018).

    Article  Google Scholar 

  190. Ahart, C. S., Blumberger, J. & Rosso, K. M. Polaronic structure of excess electrons and holes for a series of bulk iron oxides. Phys. Chem. Chem. Phys. 22, 10699–10709 (2020).

    Article  CAS  Google Scholar 

  191. Zhou, Z., Long, R. & Prezhdo, O. V. Why silicon doping accelerates electron polaron diffusion in hematite. J. Am. Chem. Soc. 141, 20222–20233 (2019).

    Article  CAS  Google Scholar 

  192. Smart, T. J., Pham, T. A., Ping, Y. & Ogitsu, T. Optical absorption induced by small polaron formation in transition metal oxides: The case of Co3O4. Phys. Rev. Mater. 3, 102401 (2019).

    Article  CAS  Google Scholar 

  193. Wu, F. & Ping, Y. Combining Landau–Zener theory and kinetic Monte Carlo sampling for small polaron mobility of doped BiVO4 from first-principles. J. Mater. Chem. A 6, 20025–20036 (2018).

    Article  CAS  Google Scholar 

  194. Ho, Q. D., Frauenheim, T. & Deak, P. Theoretical confirmation of the polaron model for the Mg acceptor in β-Ga2O3. J. Appl. Phys. 124, 145702 (2018).

    Article  Google Scholar 

  195. Deák, P. et al. Defect calculations with hybrid functionals in layered compounds and in slab models. Phys. Rev. B 100, 235304 (2019).

    Article  Google Scholar 

  196. Krüger, P. et al. Defect states at the TiO2(110) surface probed by resonant photoelectron diffraction. Phys. Rev. Lett. 100, 055501 (2008).

    Article  Google Scholar 

  197. Yim, C. M. et al. Engineering polarons at a metal oxide surface. Phys. Rev. Lett. 117, 116402 (2016).

    Article  CAS  Google Scholar 

  198. Furubayashi, Y. et al. A transparent metal: Nb-doped anatase TiO2. Appl. Phys. Lett. 86, 252101 (2005).

    Article  Google Scholar 

  199. Grätzel, M. Dye-sensitized solar cells. J. Photochem. Photobiol. C 4, 143–153 (2003).

    Article  Google Scholar 

  200. Zhang, S. X. et al. Niobium doped TiO2: Intrinsic transparent metallic anatase versus highly resistive rutile phase. J. Appl. Phys. 102, 013701 (2007).

    Article  Google Scholar 

  201. Urushibara, A. et al. Insulator-metal transition and giant magnetoresistance in La1−xSrxMnO3. Phys. Rev. B 51, 14103–14109 (1995).

    Article  CAS  Google Scholar 

  202. Tuller, H. & Nowick, A. Small polaron electron transport in reduced CeO2 single crystals. J. Phys. Chem. Solids 38, 859–867 (1977).

    Article  CAS  Google Scholar 

  203. Kang, S. D., Dylla, M. & Snyder, G. J. Thermopower-conductivity relation for distinguishing transport mechanisms: polaron hopping in CeO2 and band conduction in SrTiO3. Phys. Rev. B 97, 235201 (2018).

    Article  CAS  Google Scholar 

  204. Mechelen, J. L. M. et al. Electron-phonon interaction and charge carrier mass enhancement in SrTiO3. Phys. Rev. Lett. 100, 226403 (2008).

    Article  Google Scholar 

  205. Yoon, S. et al. Raman and optical spectroscopic studies of small-to-large polaron crossover in the perovskite manganese oxides. Phys. Rev. B 58, 2795–2801 (1998).

    Article  CAS  Google Scholar 

  206. Scott, J., Damen, T., Silfvast, W., Leite, R. & Cheesman, L. Resonant Raman scattering in ZnS and ZnSe with the cadmium laser. Opt. Commun. 1, 397–399 (1970).

    Article  CAS  Google Scholar 

  207. Kuroda, N. & Nishina, Y. Resonance Raman scattering study on exciton and polaron anisotropies in InSe. Solid State Commun. 34, 481–484 (1980).

    Article  CAS  Google Scholar 

  208. Ament, L. J. P., van Veenendaal, M. & van den Brink, J. Determining the electron-phonon coupling strength from resonant inelastic X-ray scattering at transition metal L-edges. EPL 95, 27008 (2011).

    Article  Google Scholar 

  209. Rossi, M. et al. Experimental determination of momentum-resolved electron-phonon coupling. Phys. Rev. Lett. 123, 027001 (2019).

    Article  CAS  Google Scholar 

  210. Kispert, L. D., Joseph, J., Miller, G. G. & Baughman, R. H. EPR study of polarons in a conducting polymer with nondegenerate ground states: Alkali metal complexes of poly(p-phenylene) and phenylene oligomers. J. Chem. Phys. 81, 2119–2125 (1984).

    Article  CAS  Google Scholar 

  211. Sang, L., Zhao, Y. & Burda, C. TiO2 nanoparticles as functional building blocks. Chem. Rev. 114, 9283–9318 (1985).

    Article  Google Scholar 

  212. Burroughes, J. H. et al. Light-emitting diodes based on conjugated polymers. Nature 347, 539–541 (1990).

    Article  CAS  Google Scholar 

  213. Furukawa, Y., Sakamoto, A., Ohta, H. & Tasumi, M. Raman characterization of polarons, bipolarons and solitons in conducting polymers. Synth. Met. 49, 335–340 (1992).

    Article  CAS  Google Scholar 

  214. Grenier, P., Bernier, G., Jandl, S., Salce, B. & Boatner, L. A. Fluorescence and ferroelectric microregions in KtaO3. J. Phys. Condens. Matter. 1, 2515–2520 (1989).

    Article  CAS  Google Scholar 

  215. Strocov, V. N., Cancellieri, C. & Mishchenko, A. S. Electrons and Polarons at Oxide Interfaces Explored by Soft-X-Ray ARPES (Springer, 2018).

  216. Citrin, P. H., Eisenberger, P. & Hamann, D. R. Phonon broadening of X-ray photoemission linewidths. Phys. Rev. Lett. 33, 965–969 (1974).

    Article  CAS  Google Scholar 

  217. Katz, J. E. et al. Electron small polarons and their mobility in iron (oxyhydr)oxide nanoparticles. Science 337, 1200–1203 (2012).

    Article  CAS  Google Scholar 

  218. Vura-Weis, J. et al. Femtosecond M2,3-edge spectroscopy of transition-metal oxides: photoinduced oxidation state change in α-Fe2O3. J. Phys. Chem. Lett. 4, 3667–3671 (2013).

    Article  CAS  Google Scholar 

  219. Obara, Y. et al. Femtosecond time-resolved X-ray absorption spectroscopy of anatase TiO2 nanoparticles using XFEL. Struct. Dyn. 4, 044033 (2017).

    Article  Google Scholar 

  220. Grübel, G., Stephenson, G., Gutt, C., Sinn, H. & Tschentscher, T. XPCS at the European X-ray free electron laser facility. Nucl. Instrum. Methods Phys. Res. B 262, 357–367 (2007).

    Article  Google Scholar 

  221. Di Valentin, C. et al. Density functional theory and electron paramagnetic resonance study on the effect of N–F codoping of TiO2. Chem. Mater. 20, 3706–3714 (2008).

    Article  Google Scholar 

  222. Possenriede, E., Kröse, H., Varnhorst, T., Scharfschwerdt, R. & Schirmer, O. F. Shallow acceptor and electron conduction states in BaTiO3. Ferroelectrics 151, 199–204 (1994).

    Article  CAS  Google Scholar 

  223. Chiesa, M., Paganini, M. C., Livraghi, S. & Giamello, E. Charge trapping in TiO2 polymorphs as seen by electron paramagnetic resonance spectroscopy. Phys. Chem. Chem. Phys. 15, 9435–9447 (2013).

    Article  CAS  Google Scholar 

  224. Shengelaya, A., Zhao, G.-m, Keller, H. & Müller, K. A. EPR evidence of Jahn-Teller polaron formation in La1−xCaxMnO3+y. Phys. Rev. Lett. 77, 5296–5299 (1996).

    Article  CAS  Google Scholar 

  225. Allodi, G., Cestelli Guidi, M., De Renzi, R., Caneiro, A. & Pinsard, L. Ultraslow polaron dynamics in low-doped manganites from 139La NMR-NQR and muon spin rotation. Phys. Rev. Lett. 87, 127206 (2001).

    Article  CAS  Google Scholar 

  226. Wu, L., Klie, R. F., Zhu, Y. & Jooss, C. Experimental confirmation of Zener-polaron-type charge and orbital ordering in Pr1−xCaxMnO3. Phys. Rev. B 76, 174210 (2007).

    Article  Google Scholar 

  227. Esch, F. et al. Electron localization determines defect formation on ceria substrates. Science 309, 752–755 (2005).

    Article  CAS  Google Scholar 

  228. Minato, T. et al. The electronic structure of oxygen atom vacancy and hydroxyl impurity defects on titanium dioxide (110) surface. J. Chem. Phys. 130, 124502 (2009).

    Article  Google Scholar 

  229. Giessibl, F. J. The qPlus sensor, a powerful core for the atomic force microscope. Rev. Sci. Instrum. 90, 011101 (2019).

    Article  Google Scholar 

  230. Setvin, M. et al. Polarity compensation mechanisms on the perovskite surface KTaO3(001). Science 359, 572–575 (2018).

    Article  CAS  Google Scholar 

  231. Gross, L., Mohn, F., Liljeroth, P., Giessibl, F. J. & Meyer, G. Measuring the charge state of an adatom with noncontact atomic force microscopy. Science 324, 1428–1431 (2009).

    Article  CAS  Google Scholar 

  232. Patera, L., Queck, F., Scheuerer, P. & Repp, J. Mapping orbital changes upon electron transfer with tunnelling microscopy on insulators. Nature 566, 245–248 (2019).

    Article  CAS  Google Scholar 

  233. Wagner, C. et al. Quantitative imaging of electric surface potentials with single-atom sensitivity. Nat. Mater. 18, 853–859 (2019).

    Article  CAS  Google Scholar 

  234. Dohnálek, Z., Lyubinetsky, I. & Rousseau, R. Thermally-driven processes on rutile TiO2(110)-(1×1): A direct view at the atomic scale. Prog. Surf. Sci. 85, 161–205 (2010).

    Article  Google Scholar 

  235. Liu, B. et al. Intrinsic intermediate gap states of TiO2 materials and their roles in charge carrier kinetics. J. Photochem. Photobiol. C 39, 1–57 (2019).

    Article  Google Scholar 

  236. Rousseau, R., Glezakou, V.-A. & Selloni, A. Theoretical insights into the surface physics and chemistry of redox-active oxides. Nat. Rev. Mater. 5, 460–475 (2020).

    Article  CAS  Google Scholar 

  237. Garcia, J. C., Nolan, M., Aaron Deskins, N. & Deskins, N. A. The nature of interfaces and charge trapping sites in photocatalytic mixed-phase TiO2 from first principles modeling. J. Chem. Phys. 142, 024708 (2015).

    Article  Google Scholar 

  238. Kullgren, J., Huy, H. A., Aradi, B., Frauenheim, T. & Deak, P. Theoretical study of charge separation at the rutile–anatase interface. Phys. Status Solidi Rapid Res. Lett. 8, 566–570 (2014).

    Article  CAS  Google Scholar 

  239. Carey, J. J. & McKenna, K. P. Screening doping strategies to mitigate electron trapping at anatase TiO2 surfaces. J. Phys. Chem. C 123, 22358–22367 (2019).

    Article  CAS  Google Scholar 

  240. Wang, Y. et al. Role of point defects on the reactivity of reconstructed anatase titanium dioxide (001) surface. Nat. Commun. 4, 2214 (2013).

    Article  Google Scholar 

  241. Sokolović, I. et al. Resolving the adsorption of molecular O2 on the rutile TiO2 (110) surface by noncontact atomic force microscopy. Proc. Natl Acad. Sci. USA 117, 14827–14837 (2020).

    Article  Google Scholar 

  242. Zhang, Q. et al. Measurement and manipulation of the charge state of an adsorbed oxygen adatom on the rutile TiO2(110)-1×1 surface by nc-AFM and KPFM. J. Am. Chem. Soc. 140, 15668–15674 (2018).

    Article  CAS  Google Scholar 

  243. Cao, Y. et al. Scenarios of polaron-involved molecular adsorption on reduced TiO2(110) surfaces. Sci. Rep. 7, 6148 (2017).

    Article  Google Scholar 

  244. Cao, Y. et al. Nitric oxide reaction pathways on rutile TiO2(110): the influence of surface defects and reconstructions. J. Phys. Chem. C 122, 23441–23450 (2018).

    Article  CAS  Google Scholar 

  245. Xu, M. et al. The surface science approach for understanding reactions on oxide powders: the importance of IR spectroscopy. Angew. Chem. Int. Ed. 51, 4731–4734 (2012).

    Article  CAS  Google Scholar 

  246. Kunat, M. et al. Formation of weakly bound, ordered adlayers of CO on rutile TiO2(110): a combined experimental and theoretical study. J. Chem. Phys. 130, 144703 (2009).

    Article  CAS  Google Scholar 

  247. Zhao, Y. et al. What are the adsorption sites for CO on the reduced TiO2(110)-1×1 surface? J. Am. Chem. Soc. 131, 7958–7959 (2009).

    Article  CAS  Google Scholar 

  248. Yoon, Y. et al. Anticorrelation between surface and subsurface point defects and the impact on the redox chemistry of TiO2(110). ChemPhysChem 16, 313–321 (2015).

    Article  CAS  Google Scholar 

  249. Yu, Y. Y. & Gong, X. Q. CO oxidation at rutile TiO2(110): role of oxygen vacancies and titanium interstitials. ACS Catal. 5, 2042–2050 (2015).

    Article  CAS  Google Scholar 

  250. Mu, R. et al. Adsorption and photodesorption of CO from charged point defects on TiO2(110). J. Phys. Chem. Lett. 8, 4565–4572 (2017).

    Article  CAS  Google Scholar 

  251. Chen, J., Penschke, C., Alavi, A. & Michaelides, A. Small polarons and the Janus nature of TiO2(110). Phys. Rev. B 101, 115402 (2020).

    Article  CAS  Google Scholar 

  252. Yim, C. M. et al. Visualization of water-induced surface segregation of polarons on rutile TiO2(110). J. Phys. Chem. Lett. 9, 4865–4871 (2018).

    Article  CAS  Google Scholar 

  253. López-Caballero, P. et al. Exploring the properties of Ag5–TiO2 interfaces: stable surface polaron formation, UV-Vis optical response, and CO2 photoactivation. J. Mater. Chem. A 8, 6842–6853 (2020).

    Article  Google Scholar 

  254. Selli, D., Fazio, G. & Di Valentin, C. Using density functional theory to model realistic TiO2 nanoparticles, their photoactivation and interaction with water. Catalysts 7, 357 (2017).

    Article  Google Scholar 

  255. Shirai, K. et al. Water-assisted hole trapping at the highly curved surface of nano-TiO2 photocatalyst. J. Am. Chem. Soc. 140, 1415–1422 (2018).

    Article  CAS  Google Scholar 

  256. Gerritson, H. J. in Proceedings of the First International Conference on Paramagnetic Resonance Vol. 1 (Academic, 1962).

  257. Bogomolov, V. N., Kudinov, E. K., Mirlin, D. N. & Firsov, Y. A. Polaron mechanism of light absorption in rutile crystals TiO2. Fiz. Tverd. Tela 9, 2077 (1967).

    CAS  Google Scholar 

  258. Bogomolov, V. N. & Mirlin, D. N. Optical absorption by polarons in rutile (TiO2) single crystals. Phys. Status Solidi B 27, 443–453 (1968).

    Article  CAS  Google Scholar 

  259. Dominik, L. A. K. & MacCrone, R. K. Dielectric relaxation of hopping electrons in reduced rutile, TiO2. Phys. Rev. 156, 910–913 (1967).

    Article  CAS  Google Scholar 

  260. Yagi, E., Hasiguti, R. R. & Aono, M. Electronic conduction above 4 K of slightly reduced oxygen-deficient rutile TiO2−x. Phys. Rev. B 54, 7945–7956 (1996).

    Article  CAS  Google Scholar 

  261. Nowotny, J., Radecka, M. & Rekas, M. Semiconducting properties of undoped TiO2. J. Phys. Chem. Solids 58, 927–937 (1997).

    Article  CAS  Google Scholar 

  262. Finazzi, E., Valentin, C. D. & Pacchioni, G. Nature of Ti interstitials in reduced bulk anatase and rutile TiO2. J. Phys. Chem. C 113, 3382–3385 (2009).

    Article  CAS  Google Scholar 

  263. Deák, P., Aradi, B. & Frauenheim, T. Polaronic effects in TiO2 calculated by the HSE06 hybrid functional: Dopant passivation by carrier self-trapping. Phys. Rev. B 83, 155207 (2011).

    Article  Google Scholar 

  264. Morgan, B. J., Scanlon, D. O. & Watson, G. W. Small polarons in Nb- and Ta-doped rutile and anatase TiO2. J. Mater. Chem. 19, 5175–5178 (2009).

    Article  CAS  Google Scholar 

  265. Pham, T. D. & Deskins, N. A. Efficient method for modeling polarons using electronic structure methods. J. Chem. Theory Comput. 16, 5264–5278 (2020).

    Article  CAS  Google Scholar 

  266. Deskins, N. A., Rousseau, R. & Dupuis, M. Localized electronic states from surface hydroxyls and polarons in TiO2(110). J. Phys. Chem. C 113, 14583–14586 (2009).

    Article  CAS  Google Scholar 

  267. Krüger, P. et al. Intrinsic nature of the excess electron distribution at the TiO2(110) surface. Phys. Rev. Lett. 108, 126803 (2012).

    Article  Google Scholar 

  268. Morita, K., Shibuya, T. & Yasuoka, K. Stability of excess electrons introduced by Ti interstitial in rutile TiO2(110) surface. J. Phys. Chem. C 121, 1602–1607 (2017).

    Article  CAS  Google Scholar 

  269. Moses, P. G., Janotti, A., Franchini, C., Kresse, G. & Van De Walle, C. G. Donor defects and small polarons on the TiO2(110) surface. J. Appl. Phys. 119, 181503 (2016).

    Article  Google Scholar 

  270. Deák, P., Aradi, B. & Frauenheim, T. Oxygen deficiency in TiO2: Similarities and differences between the Ti self-interstitial and the O vacancy in bulk rutile and anatase. Phys. Rev. B 92, 045204 (2015).

    Article  Google Scholar 

  271. Nelson, J., Haque, S. A., Klug, D. R. & Durrant, J. R. Trap-limited recombination in dye-sensitized nanocrystalline metal oxide electrodes. Phys. Rev. B 63, 205321 (2001).

    Article  Google Scholar 

  272. Nelson, J. & Chandler, R. E. Random walk models of charge transfer and transport in dye sensitized systems. Coord. Chem. Rev. 248, 1181–1194 (2004).

    Article  CAS  Google Scholar 

  273. Barzykin, A. V. & Tachiya, M. Mechanism of charge recombination in dye-sensitized nanocrystalline semiconductors: random flight model. J. Phys. Chem. B 106, 4356–4363 (2002).

    Article  CAS  Google Scholar 

  274. Carey, J. J. & McKenna, K. P. Does polaronic self-trapping occur at anatase TiO2 surfaces? J. Phys. Chem. C 122, 27540–27553 (2018).

    Article  CAS  Google Scholar 

  275. Deak, P., Kullgren, J. & Frauenheim, T. Polarons and oxygen vacancies at the surface of anatase TiO2. Phys. Status Solidi Rapid Res. Lett. 8, 583–586 (2014).

    Article  CAS  Google Scholar 

  276. Morgan, B. J. & Watson, G. W. Intrinsic n-type defect formation in TiO2: a comparison of rutile and anatase from GGA+U calculations. J. Phys. Chem. C 114, 2321–2328 (2010).

    Article  CAS  Google Scholar 

  277. Di Liberto, G., Tosoni, S. & Pacchioni, G. Nitrogen doping in coexposed (001)–(101) anatase TiO2 surfaces: a DFT study. Phys. Chem. Chem. Phys. 21, 21497–21505 (2019).

    Article  Google Scholar 

  278. Chiesa, M., Livraghi, S., Giamello, E., Albanese, E. & Pacchioni, G. Ferromagnetic interactions in highly stable, partially reduced TiO2: the S=2 state in anatase. Angew. Chem. Int. Ed. 56, 2604–2607 (2017).

    Article  CAS  Google Scholar 

  279. Henrich, V. E., Dresselhaus, G. & Zeiger, H. J. Observation of two-dimensional phases associated with defect states on the surface of TiO2. Phys. Rev. Lett. 36, 1335–1339 (1976).

    Article  CAS  Google Scholar 

  280. Gionco, C. et al. Al- and Ga-doped TiO2, ZrO2, and HfO2: the nature of O 2p trapped holes from a combined electron paramagnetic resonance (EPR) and density functional theory (DFT) study. Chem. Mater. 27, 3936–3945 (2015).

    Article  CAS  Google Scholar 

  281. Livraghi, S., Maurelli, S., Paganini, M. C., Chiesa, M. & Giamello, E. Probing the local environment of Ti3+ ions in TiO2 (rutile) by 17O. Angew. Chem. Int. Ed. 50, 8038–8040 (2011).

    Article  CAS  Google Scholar 

  282. Forro, L. et al. High mobility n-type charge carriers in large single crystals of anatase (TiO2). J. Appl. Phys. 75, 633–635 (1994).

    Article  CAS  Google Scholar 

  283. Dou, M. & Persson, C. Comparative study of rutile and anatase SnO2 and TiO2: Band-edge structures, dielectric functions, and polaron effects. J. Appl. Phys. 113, 083703 (2013).

    Article  Google Scholar 

  284. Moser, S. et al. Electron-phonon coupling in the bulk of anatase TiO2 measured by resonant inelastic X-ray spectroscopy. Phys. Rev. Lett. 115, 096404 (2015).

    Article  CAS  Google Scholar 

  285. Yan, B. et al. Anatase TiO2 — A model system for large polaron transport. ACS Appl. Mater. Interfaces 10, 38201–38208 (2018).

    Article  CAS  Google Scholar 

  286. Jena, A. K., Kulkarni, A. & Miyasaka, T. Halide perovskite photovoltaics: background, status, and future prospects. Chem. Rev. 119, 3036–3103 (2019).

    Article  CAS  Google Scholar 

  287. Welch, E., Scolfaro, L. & Zakhidov, A. Density functional theory + U modeling of polarons in organohalide lead perovskites. AIP Adv. 6, 125037 (2016).

    Article  Google Scholar 

  288. Diab, H. et al. Narrow linewidth excitonic emission in organic–inorganic lead iodide perovskite single crystals. J. Phys. Chem. Lett. 7, 5093–5100 (2016).

    Article  CAS  Google Scholar 

  289. Bokdam, M. et al. Role of polar phonons in the photo excited state of metal halide perovskites. Sci. Rep. 6, 28618 (2016).

    Article  CAS  Google Scholar 

  290. Zhu, X.-Y. & Podzorov, V. Charge carriers in hybrid organic–inorganic lead halide perovskites might be protected as large polarons. J. Phys. Chem. Lett. 6, 4758–4761 (2015).

    Article  CAS  Google Scholar 

  291. Emin, D. Barrier to recombination of oppositely charged large polarons. J. Appl. Phys. 123, 055105 (2018).

    Article  Google Scholar 

  292. Mahata, A., Meggiolaro, D. & De Angelis, F. From large to small polarons in lead, tin, and mixed lead–tin halide perovskites. J. Phys. Chem. Lett. 10, 1790–1798 (2019).

    Article  CAS  Google Scholar 

  293. Zhou, L. et al. Cation alloying delocalizes polarons in lead halide perovskites. J. Phys. Chem. Lett. 10, 3516–3524 (2019).

    Article  CAS  Google Scholar 

  294. Neukirch, A. J. et al. Geometry distortion and small polaron binding energy changes with ionic substitution in halide perovskites. J. Phys. Chem. Lett. 9, 7130–7136 (2018).

    Article  CAS  Google Scholar 

  295. Schmidt, J., Marques, M. R. G., Botti, S. & Marques, M. A. L. Recent advances and applications of machine learning in solid-state materials science. npj Comput. Mater. 5, 83 (2019).

    Article  Google Scholar 

  296. Wang, C.-I., Braza, M. K. E., Claudio, G. C., Nellas, R. B. & Hsu, C.-P. Machine learning for predicting electron transfer coupling. J. Phys. Chem. A 123, 7792–7802 (2019).

    Article  CAS  Google Scholar 

  297. Birschitzky, V., Reticcioli, M. & Franchini, C. Polaron Configurational Energies using Machine Learning. Master’s thesis, Univ. Vienna (2020).

  298. Behler, J. & Parrinello, M. Generalized neural-network representation of high-dimensional potential-energy surfaces. Phys. Rev. Lett. 98, 146401 (2007).

    Article  Google Scholar 

  299. Westermayr, J., Faber, F. A., Christensen, A. S., von Lilienfeld, O. A. & Marquetand, J. Neural networks and kernel ridge regression for excited states dynamics of CH2NH2+: From single-state to multi-state representations and multi-property machine learning models. Mach. Learn. Sci. Technol. 1, 025009 (2020).

    Article  Google Scholar 

  300. Jinnouchi, R., Karsai, F. & Kresse, G. On-the-fly machine learning force field generation: Application to melting points. Phys. Rev. B 100, 014105 (2019).

    Article  CAS  Google Scholar 

  301. Laanait, N., Ziatdinov, M., He, Q. & Borisevich, A. Identifying local structural states in atomic imaging by computer vision. Adv. Struct. Chem. Imaging 2, 14 (2016).

    Article  Google Scholar 

  302. Ziatdinov, M. et al. Deep analytics of atomically-resolved images: manifest and latent features. Preprint at arXiv https://arxiv.org/abs/1801.05133 (2018).

  303. Pielmeier, F. & Giessibl, F. J. Spin resolution and evidence for superexchange on NiO(001) observed by force microscopy. Phys. Rev. Lett. 110, 266101 (2013).

    Article  Google Scholar 

  304. Tan, G., Zhao, L.-D. & Kanatzidis, M. G. Rationally designing high-performance bulk thermoelectric materials. Chem. Rev. 116, 12123–12149 (2016).

    Article  CAS  Google Scholar 

  305. Biswas, S., Wallentine, S., Bandaranayake, S. & Baker, L. R. Controlling polaron formation at hematite surfaces by molecular functionalization probed by XUV reflection-absorption spectroscopy. J. Chem. Phys. 151, 104701 (2019).

    Article  Google Scholar 

  306. Pekar, S. I. Theory of colored crystals. Zh. Eksp. Teor. Fiz. 17, 868 (1947).

    CAS  Google Scholar 

  307. Pekar, S. I. New view on electronic conductivity of ionic crystals. Zh. Eksp. Teor. Fiz. 18, 105 (1948).

    CAS  Google Scholar 

  308. Landau, L. D. & Pekar, S. I. Effective mass of a polaron. Zh. Eksp. Teor. Fiz. 18, 419 (1948).

    CAS  Google Scholar 

  309. Yamashita, J. & Kurosawa, T. On electronic current in NiO. J. Phys. Chem. Solids 5, 34–43 (1958).

    Article  CAS  Google Scholar 

  310. Sewell, G. L. Electrons in polar crystals. Philos. Mag. 3, 1361 (1958).

    Article  CAS  Google Scholar 

  311. Lang, I. G. & Firsov, Y. A. Mobility of small-radius polarons at low temperatures. Zh. Eksp. Teor. Fiz. 45, 378 (1964).

    Google Scholar 

  312. Mackrodt, W., Simson, E.-A. & Harrison, N. An ab initio Hartree-Fock study of the electron-excess gap states in oxygen-deficient rutile TiO2. Surf. Sci. 384, 192–200 (1997).

    Article  CAS  Google Scholar 

  313. Franchini, C., Kresse, G. & Podloucky, R. Polaronic hole trapping in doped BaBiO3. Phys. Rev. Lett. 102, 256402 (2009).

    Article  CAS  Google Scholar 

  314. Frederikse, H. P. R., Thurber, W. R. & Hosler, W. R. Electronic transport in strontium titanate. Phys. Rev. 134, A442–A445 (1964).

    Article  Google Scholar 

  315. Lascaray, J., Desfours, J. & Averous, M. Bound magnetic polaron evidence in EuO. Solid State Commun. 19, 677–679 (1976).

    Article  CAS  Google Scholar 

  316. Friend, R. H., Bradley, D. D. C. & Townsend, P. D. Photo-excitation in conjugated polymers. J. Phys. D 20, 1367–1384 (1987).

    Article  CAS  Google Scholar 

  317. Freytag, F., Corradi, G. & Imlau, M. Atomic insight to lattice distortions caused by carrier self-trapping in oxide materials. Sci. Rep. 6, 36929 (2016).

    Article  CAS  Google Scholar 

  318. Devreese, J. T. Polarons. Encycl. Appl. Phys. 14, 383–409 (1996).

    Google Scholar 

  319. Schirmer, O. EPR Investigations of Small Electron and Hole Polarons in Oxide Perovskites (Springer, 2010).

  320. Alexandrov, A. S. & Bratkovsky, A. M. Alexandrov and Bratkovsky reply. Phys. Rev. Lett. 84, 2043 (2000).

    Article  CAS  Google Scholar 

  321. Alexandrov, A. S. & Mott, N. F. Polarons and bipolarons (World Scientific, 1996).

  322. Lanzara, A. et al. Evidence for ubiquitous strong electron–phonon coupling in high-temperature superconductors. Nature 412, 510–514 (2001).

    Article  CAS  Google Scholar 

  323. Nishio, T., Ahmad, J. & Uwe, H. Spectroscopic observation of bipolaronic point defects in Ba1−xKxBiO3. Phys. Rev. Lett. 95, 176403 (2005).

    Article  Google Scholar 

  324. Ahmad, J. & Uwe, H. Small-polaron excitations in Ba1−xKxBiO3 studied by optical reflectivity measurements. Phys. Rev. B 72, 125103 (2005).

    Article  Google Scholar 

  325. de Gennes, P. G. Effects of double exchange in magnetic crystals. Phys. Rev. 118, 141–154 (1960).

    Article  Google Scholar 

  326. Mauger, A. Magnetic polaron: Theory and experiment. Phys. Rev. B 27, 2308–2324 (1983).

    Article  CAS  Google Scholar 

  327. Umehara, M. Density-functional approach to doped magnetic semiconductors: evolution of bound states of electrons as the donor concentration increases. Phys. Rev. B 41, 2421–2433 (1990).

    Article  CAS  Google Scholar 

  328. Bondarenko, N. et al. Spin polaronics: Static and dynamic properties of spin polarons in La-doped CaMnO3. Phys. Rev. B 100, 134443 (2019).

    Article  CAS  Google Scholar 

  329. Lenjer, S., Schirmer, O. F., Hesse, H. & Kool, T. W. Conduction states in oxide perovskites: Three manifestations of Ti3+ Jahn-Teller polarons in barium titanate. Phys. Rev. B 66, 165106 (2002).

    Article  Google Scholar 

  330. Eglitis, R. I. Ab initio calculations of SrTiO3, BaTiO3, PbTiO3, CaTiO3, SrZrO3, PbZrO3 and BaZrO3 (001), (011) and (111) surfaces as well as F centers, polarons, KTN solid solutions and Nb impurities therein. Int. J. Mod. Phys. B 28, 1430009 (2014).

    Article  Google Scholar 

  331. Höck, K. H., Nickisch, H. & Thomas, H. Jahn-Teller effect in itinerant electron systems: the Jahn-Teller polaron. Helv. Phys. Act. 56, 237–243 (1983).

    Google Scholar 

  332. Allen, P. B. & Perebeinos, V. Anti-Jahn-Teller polaron in LaMnO3. Phys. Rev. B 60, 10747–10753 (1999).

    Article  CAS  Google Scholar 

  333. Hao, X., Wang, Z., Schmid, M., Diebold, U. & Franchini, C. Coexistence of trapped and free excess electrons in SrTiO3. Phys. Rev. B 91, 085204 (2015).

    Article  Google Scholar 

  334. Klyukin, K. & Alexandrov, V. Effect of intrinsic point defects on ferroelectric polarization behavior of SrTiO3. Phys. Rev. B 95, 035301 (2017).

    Article  Google Scholar 

  335. Zheng, G. & Patterson, C. H. Ferromagnetic polarons in La0.5Ca0.5MnO3 and La0.33Ca0.67MnO3. Phys. Rev. B 67, 220404 (2003).

    Article  Google Scholar 

  336. Giovannetti, G., Kumar, S., van den Brink, J. & Picozzi, S. Magnetically induced electronic ferroelectricity in half-doped manganites. Phys. Rev. Lett. 103, 037601 (2009).

    Article  Google Scholar 

  337. Bao, E. & L. Xiao, J. Temperature dependence of the properties of the strong-coupling polaron in a slab of polar crystal. J. Phys. Soc. Jpn. 72, 627–633 (2003).

    Article  CAS  Google Scholar 

  338. Toyozawa, Y. Interband effect of lattice vibrations in the exciton absorption spectra. J. Phys. Chem. Solids 25, 59–71 (1964).

    Article  CAS  Google Scholar 

  339. Nagaev, E. L. Exciton polarons and plasma-exciton waves. Zh. Eksp. Teor. Fiz. 57, 469 (1969).

    CAS  Google Scholar 

  340. Iadonisi, G. & Bassani, F. Polaronic correction to the exciton effective mass. Il Nuovo Cim. D. 9, 703–714 (1987).

    Article  Google Scholar 

  341. Hsu, H.-C. & Hsieh, W.-F. Excitonic polaron and phonon assisted photoluminescence of ZnO nanowires. Solid State Commun. 131, 371–375 (2004).

    Article  CAS  Google Scholar 

  342. Mahrt, R. & Bassler, H. Light and heavy excitonic polarons in conjugated polymers. Synth. Met. 45, 107–117 (1991).

    Article  CAS  Google Scholar 

  343. Mott, N. F. & Stoneham, A. M. The lifetime of electrons, holes and excitons before self-trapping. J. Phys. C Solid State Phys. 10, 3391–3398 (1977).

    Article  CAS  Google Scholar 

  344. Schein, L. & Borsenberger, P. Hole mobilities in a hydrazone-doped polycarbonate and poly(styrene). Chem. Phys. 177, 773–781 (1993).

    Article  CAS  Google Scholar 

  345. Janotti, A., Varley, J. B., Choi, M. & Van de Walle, C. G. Vacancies and small polarons in SrTiO3. Phys. Rev. B 90, 085202 (2014).

    Article  CAS  Google Scholar 

  346. Diebold, U., Li, S.-C. & Schmid, M. Oxide surface science. Annu. Rev. Phys. Chem. 61, 129–148 (2010).

    Article  CAS  Google Scholar 

Download references

Acknowledgements

We thank the Erwin Schrödinger Institute (ESI) for hosting the ESI-CECAM workshop “Polarons in the 21st Century” and all participants for the many enlightening discussions. C.F. acknowledges support by the Austrian Science Fund (FWF) project nos. I 2460 and I 4506, and the many enriching discussions with the project partners J. Tempere, G. Kresse, S. Klimin and J. Devreese. M.S. acknowledges support by the Czech Science Foundation (GACR 20-21727X) and GAUK Primus/20/SCI/009. U.D. acknowledges support by the Austrian Science Fund (FWF) (Wittgenstein Prize Z-250).

Author information

Authors and Affiliations

Authors

Contributions

All authors contributed equally to the writing of the manuscript.

Corresponding author

Correspondence to Cesare Franchini.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Franchini, C., Reticcioli, M., Setvin, M. et al. Polarons in materials. Nat Rev Mater 6, 560–586 (2021). https://doi.org/10.1038/s41578-021-00289-w

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41578-021-00289-w

This article is cited by

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing